System and method balance-point-thermal-conductivity-based building analysis the aid of a digital computer

ABSTRACT

A system and method for balance-point-thermal-conductivity-based building analysis with the aid of a digital computer are provided. A total thermal conductivity of a building is obtained. A balance point thermal conductivity of the building is identified. The balance point thermal conductivity is divided by an area of the building to obtain the balance point thermal conductivity per unit of the area. A further balance point thermal conductivity per the unit of a further area of at least one further building and a further total thermal conductivity of the at least one further building is obtained. The balance point thermal conductivity per unit of the area of the building is compared to the further balance point thermal conductivity per the unit of the further area of the at least one further building and the total thermal conductivity is compared to the further total conductivity of the at least one building.

FIELD

This application relates in general to energy conservation and, inparticular, to a system and method forbalance-point-thermal-conductivity-based building analysis with the aidof a digital computer.

BACKGROUND

Concern has been growing in recent days over energy consumption in theUnited States and abroad. The cost of energy has steadily risen as powerutilities try to cope with continually growing demand, increasing fuelprices, and stricter regulatory mandates. Power utilities must alsomaintain existing infrastructure, while simultaneously finding ways toadd more generation capacity to meet future needs, both of which add tothe cost of energy. Moreover, burgeoning energy consumption continues toimpact the environment and deplete natural resources.

A large proportion of the rising cost of energy is borne by consumers,who, despite the need, remain poorly-equipped to identify the most costeffective ways to lower their own energy consumption. Often, no-costbehavioral changes, such as adjusting thermostat settings and turningoff unused appliances, and low-cost physical improvements, such asswitching to energy-efficient light bulbs, are insufficient to offsetutility bill increases. Rather, appreciable decreases in energyconsumption are often only achievable by investing in upgrades to abuilding's heating or cooling envelope. Identifying and comparing thekinds of building shell improvements that will yield an acceptablereturn on investment in terms of costs versus likely energy savings,though, requires finding many building-specific parameters, especiallythe building's thermal conductivity (UA^(Total)).

The costs of energy for heating, ventilating, and air conditioning(HVAC) system operation are often significant contributors to utilitybills for both homeowners and businesses, and HVAC energy costs aredirectly tied to a building's thermal efficiency. For instance, a poorlyinsulated house or a building with significant sealing problems willrequire more overall HVAC usage to maintain a desired interiortemperature than would a comparably-sized but well-insulated and sealedstructure. Lowering HVAC energy costs is not as simple as choosing athermostat setting to cause an HVAC system to run for less time or lessfrequently. Rather, HVAC system efficiency, duration of heating orcooling seasons, differences between indoor and outdoor temperatures,and other factors, in addition to thermal efficiency, can weigh intooverall energy consumption.

Conventionally, estimating periodic HVAC energy consumption and fuelcosts begins with analytically determining a building's thermalconductivity UA^(Total) through an on-site energy audit. A typicalenergy audit involves measuring physical dimensions of walls, windows,doors, and other building parts; approximating R-values for thermalresistance; estimating infiltration using a blower door test; anddetecting air leakage using a thermal camera, after which a numericalmodel is run to solve for thermal conductivity. The UA^(Total) result iscombined with the duration of the heating or cooling season, asapplicable, over the period of inquiry and adjusted for HVAC systemefficiency, plus any solar savings fraction. The audit report is oftenpresented in the form of a checklist of corrective measures that may betaken to improve the building's shell and HVAC system, and thereby loweroverall energy consumption for HVAC. As an involved process, an energyaudit can be costly, time-consuming, and invasive for building ownersand occupants. Further, as a numerical result derived from a theoreticalmodel, an energy audit carries an inherent potential for inaccuracystrongly influenced by mismeasurements, data assumptions, and so forth.As well, the degree of improvement and savings attributable to variouspossible improvements is not necessarily quantified due to the widerange of variables.

Therefore, a need remains for a practical model for determining actualand potential energy consumption for the heating and cooling of abuilding.

A further need remains for an approach to quantifying improvements inenergy consumption and cost savings resulting from building shellupgrades.

SUMMARY

Fuel consumption for building heating and cooling can be calculatedthrough two practical approaches that characterize a building's thermalefficiency through empirically-measured values and readily-obtainableenergy consumption data, such as available in utility bills, therebyavoiding intrusive and time-consuming analysis with specialized testingequipment. While the discussion is herein centered on building heatingrequirements, the same principles can be applied to an analysis ofbuilding cooling requirements. The first approach can be used tocalculate annual or periodic fuel requirements. The approach requiresevaluating typical monthly utility billing data and approximations ofheating (or cooling) losses and gains.

The second approach can be used to calculate hourly (or interval) fuelrequirements. The approach includes empirically deriving threebuilding-specific parameters: thermal mass, thermal conductivity, andeffective window area. HVAC system power rating and conversion anddelivery efficiency are also parametrized. The parameters are estimatedusing short duration tests that last at most several days. Theparameters and estimated HVAC system efficiency are used to simulate atime series of indoor building temperature. In addition, the secondhourly (or interval) approach can be used to verify or explain theresults from the first annual (or periodic) approach. For instance, timeseries results can be calculated using the second approach over the spanof an entire year and compared to results determined through the firstapproach. Other uses of the two approaches and forms of comparison arepossible.

In one embodiment, a system and method forbalance-point-thermal-conductivity-based building analysis with the aidof a digital computer are provided. A total thermal conductivity of abuilding is obtained by a computer, the computer including a processorconfigured to execute code stored in a memory. A balance point thermalconductivity of the building is identified by the computer based on abalance point up to which the building can be thermally sustained usingonly internal heating period for a time period. The balance pointthermal conductivity is divided by an area of the building to obtain thebalance point thermal conductivity per unit of the area. A furtherbalance point thermal conductivity per the unit of a further area of atleast one further building and a further total thermal conductivity ofthe at least one further building are obtained by the computer. Thebalance point thermal conductivity per unit of the area of the buildingis compared by the computer to the further balance point thermalconductivity per the unit of the further area of the at least onefurther building and the total thermal conductivity is compared by thecomputer to the further total conductivity of the at least one building.

The foregoing approaches, annual (or periodic) and hourly (or interval)improve upon and compliment the standard energy audit-style methodologyof estimating heating (and cooling) fuel consumption in several ways.First, per the first approach, the equation to calculate annual fuelconsumption and its derivatives is simplified over thefully-parameterized form of the equation used in energy audit analysis,yet without loss of accuracy. Second, both approaches require parametersthat can be obtained empirically, rather than from a detailed energyaudit that requires specialized testing equipment and prescribed testconditions. Third, per the second approach, a time series of indoortemperature and fuel consumption data can be accurately generated. Theresulting fuel consumption data can then be used by economic analysistools using prices that are allowed to vary over time to quantifyeconomic impact.

Moreover, the economic value of heating (and cooling) energy savingsassociated with any building shell improvement in any building has beenshown to be independent of building type, age, occupancy, efficiencylevel, usage type, amount of internal electric gains, or amount solargains, provided that fuel has been consumed at some point for auxiliaryheating. The only information required to calculate savings includes thenumber of hours that define the winter season; average indoortemperature; average outdoor temperature; the building's HVAC systemefficiency (or coefficient of performance for heat pump systems); thearea of the existing portion of the building to be upgraded; the R-valueof the new and existing materials; and the average price of energy, thatis, heating fuel.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 2 is a graph showing, by way of example, balance point thermalconductivity.

FIG. 3 is a flow diagram showing a computer-implemented method formodeling periodic building heating energy consumption in accordance withone embodiment.

FIG. 4 is a flow diagram showing a routine for determining heating gainsfor use in the method of FIG. 3 .

FIG. 5 is a flow diagram showing a routine for balancing energy for usein the method of FIG. 3 .

FIG. 6 is a process flow diagram showing, by way of example, consumerheating energy consumption-related decision points.

FIG. 7 is a table showing, by way of example, data used to calculatethermal conductivity.

FIG. 8 is a table showing, by way of example, thermal conductivityresults for each season using the data in the table of FIG. 7 as inputsinto Equations (25) through (28).

FIG. 9 is a graph showing, by way of example, a plot of the thermalconductivity results in the table of FIG. 8 .

FIG. 10 is a graph showing, by way of example, an auxiliary heatingenergy analysis and energy consumption investment options.

FIG. 11 is a functional block diagram showing heating losses and gainsrelative to a structure.

FIG. 12 is a flow diagram showing a computer-implemented method formodeling interval building heating energy consumption in accordance witha further embodiment.

FIG. 13 is a table showing the characteristics of empirical tests usedto solve for the four unknown parameters in Equation (47).

FIG. 14 is a flow diagram showing a routine for empirically determiningbuilding- and equipment-specific parameters using short duration testsfor use in the method of FIG. 12 .

FIG. 15 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach.

FIG. 16 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the allowable indoor temperaturelimited to 2° F. above desired temperature of 68° F.

FIG. 17 is a graph showing, by way of example, a comparison of auxiliaryheating energy requirements with the size of effective window areatripled from 2.5 m² to 7.5 m².

FIG. 18 is a table showing, by way of example, test data.

FIG. 19 is a table showing, by way of example, the statistics performedon the data in the table of FIG. 18 required to calculate the three testparameters.

FIG. 20 is a graph showing, by way of example, hourly indoor (measuredand simulated) and outdoor (measured) temperatures.

FIG. 21 is a graph showing, by way of example, simulated versus measuredhourly temperature delta (indoor minus outdoor).

FIG. 22 is a block diagram showing a computer-implemented system formodeling building heating energy consumption in accordance with oneembodiment.

DETAILED DESCRIPTION

Conventional Energy Audit-Style Approach

Conventionally, estimating periodic HVAC energy consumption andtherefore fuel costs includes analytically determining a building'sthermal conductivity (UA^(Total)) based on results obtained through anon-site energy audit. For instance, J. Randolf and G. Masters, Energyfor Sustainability: Technology, Planning, Policy, pp. 247, 248, 279(2008), present a typical approach to modeling heating energyconsumption for a building, as summarized by Equations 6.23, 6.27, and7.5. The combination of these equations states that annual heating fuelconsumption Q^(Fuel) equals the product of UA^(Total), 24 hours per day,and the number of heating degree days (HDD) associated with a particularbalance point temperature Ft^(Balance Point), as adjusted for the solarsavings fraction (SSF) divided by HVAC system efficiency (η^(HVAC)):

$\begin{matrix}{Q^{Fuel} = {\left( {UA}^{Total} \right)\left( {24*{HDD}^{T^{{Balance}{Point}}}} \right)\left( {1 - {SSF}} \right)\left( \frac{1}{\eta^{HVAC}} \right)}} & (1)\end{matrix}$

such that:

$\begin{matrix}{T^{{Balance}{Point}} = {T^{{Set}{Point}} - \frac{{Internal}{Gains}}{{UA}^{Total}}}} & (2)\end{matrix}$ and $\begin{matrix}{\eta^{HVAC} = {\eta^{Furnace}\eta^{Distribution}}} & (3)\end{matrix}$

where T^(Set Point) represents the temperature setting of thethermostat, Internal Gains represents the heating gains experiencedwithin the building as a function of heat generated by internal sourcesand auxiliary heating, as further discussed infra, η^(Furnace)represents the efficiency of the furnace or heat source proper, andη^(Distribution) represents the efficiency of the duct work and heatdistribution system. For clarity, HDD^(T) ^(Balance Point) will beabbreviated to HDD^(Balance Point Temp).

A cursory inspection of Equation (1) implies that annual fuelconsumption is linearly related to a building's thermal conductivity.This implication further suggests that calculating fuel savingsassociated with building envelope or shell improvements isstraightforward. In practice, however, such calculations are notstraightforward because Equation (1) was formulated with the goal ofdetermining the fuel required to satisfy heating energy needs. As such,there are several additional factors that the equation must take intoconsideration.

First, Equation (1) needs to reflect the fuel that is required only whenindoor temperature exceeds outdoor temperature. This need led to theheating degree day (HDD) approach (or could be applied on a shorter timeinterval basis of less than one day) of calculating the differencebetween the average daily (or hourly) indoor and outdoor temperaturesand retaining only the positive values. This approach complicatesEquation (1) because the results of a non-linear term must be summed,that is, the maximum of the difference between average indoor andoutdoor temperatures and zero. Non-linear equations complicateintegration, that is, the continuous version of summation.

Second, Equation (1) includes the term Balance Point temperature(T^(Balance Point)) The goal of including the term T^(Balance Point) wasto recognize that the internal heating gains of the building effectivelylowered the number of degrees of temperature that auxiliary heatingneeded to supply relative to the temperature setting of the thermostatT^(Set Point). A balance point temperature T^(Balance Point) of 65° F.was initially selected under the assumption that 65° F. approximatelyaccounted for the internal gains. As buildings became more efficient,however, an adjustment to the balance point temperatureT^(Balance Point) was needed based on the building's thermalconductivity (UA^(Total)) and internal gains. This further complicatedEquation (1) because the equation became indirectly dependent on (andinversely related to) UA^(Total) through T^(Balance Point).

Third, Equation (1) addresses fuel consumption by auxiliary heatingsources. As a result, Equation (1) must be adjusted to account for solargains. This adjustment was accomplished using the Solar Savings Fraction(SSF). The SSF is based on the Load Collector Ratio (see Eq. 7.4 inRandolf and Masters, p. 278, cited supra, for information about theLCR). The LCR, however, is also a function of UA^(Total). As a result,the SSF is a function of UA^(Total) in a complicated, non-closed formsolution manner. Thus, the SSF further complicates calculating the fuelsavings associated with building shell improvements because the SSF isindirectly dependent on UA^(Total).

As a result, these direct and indirect dependencies significantlycomplicate calculating a change in annual fuel consumption based on achange in thermal conductivity. The difficulty is made evident by takingthe derivative of Equation (1) with respect to a change in thermalconductivity. The chain and product rules from calculus need to beemployed since HDD^(Balance Point Temp) and SSF are indirectly dependenton UA^(Total):

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = {\left\{ {{\left( {UA}^{Total} \right)\left\lbrack {{\left( {HDD}^{Ba{lance}{Point}{Temp}} \right)\left( {{- \frac{dSSF}{dLCR}}\frac{dLCR}{{dUA}^{Total}}} \right)} + {\left( {\frac{{dHD}D^{Ba{lance}{Point}{Temp}}}{{dT}^{Ba{lance}{Point}}}\frac{{dT}^{Ba{lance}{Point}}}{{dUA}^{Total}}} \right)\left( {1 - {SSF}} \right)}} \right\rbrack} + {\left( {HDD}^{{Balance}{Point}{Temp}} \right)\left( {1 - {SSF}} \right)}} \right\}\left( \frac{24}{\eta^{HVAC}} \right)}} & (4)\end{matrix}$

The result is Equation (4), which is an equation that is difficult tosolve due to the number and variety of unknown inputs that are required.

To add even further complexity to the problem of solving Equation (4),conventionally, UA^(Total) is determined analytically by performing adetailed energy audit of a building. An energy audit involves measuringphysical dimensions of walls, windows, doors, and other building parts;approximating R-values for thermal resistance; estimating infiltrationusing a blower door test; and detecting air leakage. A numerical modelis then run to perform the calculations necessary to estimate thermalconductivity. Such an energy audit can be costly, time consuming, andinvasive for building owners and occupants. Moreover, as a calculatedresult, the value estimated for UA^(Total) carries the potential forinaccuracies, as the model is strongly influenced by physicalmismeasurements or omissions, data assumptions, and so forth.

Empirically-Based Approaches to Modeling Heating Fuel Consumption

Building heating (and cooling) fuel consumption can be calculatedthrough two approaches, annual (or periodic) and hourly (or interval) tothermally characterize a building without intrusive and time-consumingtests. The first approach, as further described infra beginning withreference to FIG. 1 , requires typical monthly utility billing data andapproximations of heating (or cooling) losses and gains. The secondapproach, as further described infra beginning with reference to FIG. 11, involves empirically deriving three building-specific parameters,thermal mass, thermal conductivity, and effective window area, plus HVACsystem efficiency using short duration tests that last at most severaldays. The parameters are then used to simulate a time series of indoorbuilding temperature and of fuel consumption. While the discussionherein is centered on building heating requirements, the same principlescan be applied to an analysis of building cooling requirements.

First Approach: Annual (or Periodic) Fuel Consumption

Fundamentally, thermal conductivity is the property of a material, here,a structure, to conduct heat. FIG. 1 is a functional block diagram 10showing heating losses and gains relative to a structure 11.Inefficiencies in the shell 12 (or envelope) of a structure 11 canresult in losses in interior heating 14, whereas gains 13 in heatinggenerally originate either from sources within (or internal to) thestructure 11, including heating gains from occupants 15, gains fromoperation of electric devices 16, and solar gains 17, or from auxiliaryheating sources 18 that are specifically intended to provide heat to thestructure's interior.

In this first approach, the concepts of balance point temperatures andsolar savings fractions, per Equation (1), are eliminated. Instead,balance point temperatures and solar savings fractions are replaced withthe single concept of balance point thermal conductivity. Thissubstitution is made by separately allocating the total thermalconductivity of a building (UA^(Total)) to thermal conductivity forinternal heating gains (UA^(Balance Point)), including occupancy, heatproduced by operation of certain electric devices, and solar gains, andthermal conductivity for auxiliary heating (UA^(Auxiliary Heating)) Theend result is Equation (31), further discussed in detail infra, whicheliminates the indirect and non-linear parameter relationships inEquation (1) to UA^(Total).

The conceptual relationships embodied in Equation (31) can be describedwith the assistance of a diagram. FIG. 2 is a graph 20 showing, by wayof example, balance point thermal conductivity UA^(Balance Point), thatis, the thermal conductivity for internal heating gains. The x-axis 21represents total thermal conductivity, UA^(Total), of a building (inunits of Btu/hr-° F.). The y-axis 22 represents total heating energyconsumed to heat the building. Total thermal conductivity 21 (along thex-axis) is divided into “balance point” thermal conductivity(UA^(Balance Point)) 23 and “heating system” (or auxiliary heating)thermal conductivity (UA^(Auxiliary Heating)) 24. “Balance point”thermal conductivity 23 characterizes heating losses, which can occur,for example, due to the escape of heat through the building envelope tothe outside and by the infiltration of cold air through the buildingenvelope into the building's interior that are compensated for byinternal gains. “Heating system” thermal conductivity 24 characterizesheating gains, which reflects the heating delivered to the building'sinterior above the balance point temperature T^(Balance Point) generallyas determined by the setting of the auxiliary heating source'sthermostat or other control point.

In this approach, total heating energy 22 (along the y-axis) is dividedinto gains from internal heating 25 and gains from auxiliary heatingenergy 25. Internal heating gains are broken down into heating gainsfrom occupants 27, gains from operation of electric devices 28 in thebuilding, and solar gains 29. Sources of auxiliary heating energyinclude, for instance, natural gas furnace 30 (here, with a 56%efficiency), electric resistance heating 31 (here, with a 100%efficiency), and electric heat pump 32 (here, with a 250% efficiency).Other sources of heating losses and gains are possible.

The first approach provides an estimate of fuel consumption over a yearor other period of inquiry based on the separation of thermalconductivity into internal heating gains and auxiliary heating. FIG. 3is a flow diagram showing a computer-implemented method 40 for modelingperiodic building heating energy consumption in accordance with oneembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 22 , as a series of process or method modules orsteps.

In the first part of the approach (steps 41-43), heating losses andheating gains are separately analyzed. In the second part of theapproach (steps 44-46), the portion of the heating gains that need to beprovided by fuel, that is, through the consumption of energy forgenerating heating using auxiliary heating 18 (shown in FIG. 1 ), isdetermined to yield a value for annual (or periodic) fuel consumption.Each of the steps will now be described in detail.

Specify Time Period

Heating requirements are concentrated during the winter months, so as aninitial step, the time period of inquiry is specified (step 41). Theheating degree day approach (HDD) in Equation (1) requires examining allof the days of the year and including only those days where outdoortemperatures are less than a certain balance point temperature. However,this approach specifies the time period of inquiry as the winter seasonand considers all of the days (or all of the hours, or other time units)during the winter season. Other periods of inquiry are also possible,such as a five- or ten-year time frame, as well as shorter time periods,such as one- or two-month intervals.

Separate Heating Losses from Heating Gains

Heating losses are considered separately from heating gains (step 42).The rationale for drawing this distinction will now be discussed.

Heating Losses

For the sake of discussion herein, those regions located mainly in thelower latitudes, where outdoor temperatures remain fairly moderate yearround, will be ignored and focus placed instead on those regions thatexperience seasonal shifts of weather and climate. Under thisassumption, a heating degree day (HDD) approach specifies that outdoortemperature must be less than indoor temperature. No such limitation isapplied in this present approach. Heating losses are negative if outdoortemperature exceeds indoor temperature, which indicates that thebuilding will gain heat during these times. Since the time period hasbeen limited to only the winter season, there will likely to be alimited number of days when that situation could occur and, in thoselimited times, the building will benefit by positive heating gain. (Notethat an adjustment would be required if the building took advantage ofthe benefit of higher outdoor temperatures by circulating outdoor airinside when this condition occurs. This adjustment could be made bytreating the condition as an additional source of heating gain.)

As a result, fuel consumption for heating losses Q^(Losses) over thewinter season equals the product of the building's total thermalconductivity UA^(Total) and the difference between the indoor T^(Indoor)and outdoor temperature T^(Outdoor), summed over all of the hours of thewinter season:

$\begin{matrix}{Q^{Losses} = {\sum\limits_{t^{Start}}^{t^{End}}{\left( {UA}^{Total} \right)\left( {T_{t}^{Indoor} - T_{t}^{Outdoor}} \right)}}} & (5)\end{matrix}$

where Start and End respectively represent the first and last hours ofthe winter (heating) season.

Equation (5) can be simplified by solving the summation. Thus, totalheating losses Q^(Losses) equal the product of thermal conductivityUA^(Total) and the difference between average indoor temperature T^(Indoor) and average outdoor temperature T ^(Outdoor) over the winterseason and the number of hours H in the season over which the average iscalculated:

Q ^(Losses)=(UA ^(Total))( T ^(Indoor) −T ^(Outdoor))(H)  (6)

Heating Gains

Heating gains are calculated for two broad categories (step 43) based onthe source of heating, internal heating gains Q^(Gains-Internal) andauxiliary heating gains Q^(Gains-Auxiliary Heating) as further describedinfra with reference to FIG. 4 . Internal heating gains can besubdivided into heating gained from occupants Q^(Gains-Occupants),heating gained from the operation of electric devices Q^(Gains-Electric)and heating gained from solar heating Q^(Gains-Solar). Other sources ofinternal heating gains are possible. The total amount of heating gainedQ^(Gains) from these two categories of heating sources equals:

Q ^(Gains) =Q ^(Gains-Internal) +Q ^(Gains-Auxiliary Heating)  (7)

Where

Q ^(Gains-Internal) =Q ^(Gains-Occupants) +Q ^(Gains-Electric) +Q^(Gains-Solar)  (8)

Calculate Heating Gains

Equation (8) states that internal heating gains Q^(Gains-Internal)include heating gains from Occupant, Electric, and Solar heatingsources. FIG. 4 is a flow diagram showing a routine 50 for determiningheating gains for use in the method 40 of FIG. 3 Each of these heatinggain sources will now be discussed.

Occupant Heating Gains

People occupying a building generate heat. Occupant heating gainsQ^(Gains-Occupants) (step 51) equal the product of the heat produced perperson, the average number of people in a building over the time period,and the number of hours (H) (or other time units) in that time period.Let P represent the average number of people. If a person produces 250Btu of heat per hour, heating gains from the occupantsQ^(Gains-Occupants) equal:

Q ^(Gains-Occupants)=250( P )(H)  (9)

Electric Heating Gains

The operation of electric devices that deliver all heat that isgenerated into the interior of the building, for instance, lights,refrigerators, and the like, contribute to internal heating gain.Electric heating gains Q^(Gains-Electric) (step 52) equal the amount ofelectricity used in the building that is converted to heat over the timeperiod.

Care needs to be taken to ensure that the measured electricityconsumption corresponds to the indoor usage. Two adjustments may berequired. First, many electric utilities measure net electricityconsumption. The energy produced by any photovoltaic (PV) system needsto be added back to net energy consumption (Net) to result in grossconsumption if the building has a net-metered PV system. This amount canbe estimated using time- and location-correlated solar resource data, aswell as specific information about the orientation and othercharacteristics of the photovoltaic system, such as can be provided bythe Solar Anywhere SystemCheck service (http://www.SolarAnywhere.com), aWeb-based service operated by Clean Power Research, L.L.C., Napa,Calif., with the approach described, for instance, in commonly-assignedU.S. patent application, entitled “Computer-Implemented System andMethod for Estimating Gross Energy Load of a Building,” Ser. No.14/531,940, filed Nov. 3, 2014, pending, the disclosure of which isincorporated by reference, or measured directly.

Second, some uses of electricity may not contribute heat to the interiorof the building and need be factored out as external electric heatinggains (External). These uses include electricity used for electricvehicle charging, electric dryers (assuming that most of the hot exhaustair is vented outside of the building, as typically required by buildingcode), outdoor pool pumps, and electric water heating using eitherdirect heating or heat pump technologies (assuming that most of the hotwater goes down the drain and outside the building—a large body ofstanding hot water, such as a bathtub filled with hot water, can beconsidered transient and not likely to appreciably increase thetemperature indoors over the long run).

Including the conversion factor from kWh to Btu (sinceQ^(Gains-Electric) is in units of Btu), internal electric gainsQ^(Gains-Electric) equal:

$\begin{matrix}{Q^{{Gains} - {Electric}} = {\left( \overset{\_}{{Net} + {PV} - {External}} \right)(H)\left( \frac{3,412{Btu}}{kWh} \right)}} & (10)\end{matrix}$

where Net represents net energy consumption, PV represents any energyproduced by a PV system, External represents heating gains attributableto electric sources that do not contribute heat to the interior of abuilding. The average delivered electricity Net+PV−External equals thetotal over the time period divided by the number of hours (H) in thattime period.

$\begin{matrix}{\overset{\_}{{Net} + {PV} - {External}} = \frac{{Net} + {PV} - {External}}{H}} & (11)\end{matrix}$

Solar Heating Gains

Solar energy that enters through windows, doors, and other openings in abuilding as sunlight will heat the interior. Solar heating gainsQ^(Gains-Solar) (step 53) equal the amount of heat delivered to abuilding from the sun. In the northern hemisphere, Q^(Gains-Solar) canbe estimated based on the south-facing window area (m²) times the solarheating gain coefficient (SHGC) times a shading factor; together, theseterms are represented by the effective window area (W). Solar heatinggains Q^(Gains-Solar) equal the product of W, the average directvertical irradiance (DVI) available on a south-facing surface (Solar, asrepresented by DVI in kW/m²), and the number of hours (H) in the timeperiod. Including the conversion factor from kWh to Btu (sinceQ^(Gains-Solar) is in units of Btu while average solar is in kW/m²),solar heating gains Q^(Gains-Solar) equal:

$\begin{matrix}{Q^{{Gains} - {Solar}} = {\left( \overset{\_}{Solar} \right)(W)(H)\left( \frac{3,412{Btu}}{kWh} \right)}} & (12)\end{matrix}$

Note that for reference purposes, the SHGC for one particular highquality window designed for solar gains, the Andersen High-PerformanceLow-E4 PassiveSun Glass window product, manufactured by AndersenCorporation, Bayport, Minn., is 0.54; many windows have SHGCs that arebetween 0.20 to 0.25.

Auxiliary Heating Gains

The internal sources of heating gain share the common characteristic ofnot being operated for the sole purpose of heating a building, yetnevertheless making some measureable contribution to the heat to theinterior of a building. The fourth type of heating gain, auxiliaryheating gains Q^(Gains-Auxiliary Heating) consumes fuel specifically toprovide heat to the building's interior and, as a result, must includeconversion efficiency. The gains from auxiliary heating gainsQ^(Gains-Auxiliary Heating) (step 53) equal the product of the averagehourly fuel consumed Q ^(Fuel) times the hours (H) in the period andHVAC system efficiency η^(HVAC):

Q ^(Gains-Auxiliary Heating)=( Q ^(Fuel))(H)η^(HVAC)  (13)

Divide Thermal Conductivity into Parts

Referring back to FIG. 3 , a building's thermal conductivity UA^(Total),rather than being treated as a single value, can be conceptually dividedinto two parts (step 44), with a portion of UA^(Total) allocated to“balance point thermal conductivity” (UA^(Balance Point)) and a portionto “auxiliary heating thermal conductivity” (UA^(Auxiliary Heating)),such as pictorially described supra with reference to FIG. 2 .UA^(Balance Point) corresponds to the heating losses that a building cansustain using only internal heating gains Q^(Gains-Internal). This valueis related to the concept that a building can sustain a specifiedbalance point temperature in light of internal gains. However, insteadof having a balance point temperature, some portion of the buildingUA^(Balance Point) is considered to be thermally sustainable givenheating gains from internal heating sources (Q^(Gains-Internal)). As therest of the heating losses must be made up by auxiliary heating gains,the remaining portion of the building UA^(Auxiliary Heating) isconsidered to be thermally sustainable given heating gains fromauxiliary heating sources (Q^(Gains-Auxiliary Heating)). The amount ofauxiliary heating gained is determined by the setting of the auxiliaryheating source's thermostat or other control point. Thus, UA^(Total) canbe expressed as:

UA ^(Total) =UA ^(Balance Point) +UA ^(Auxiliary Heating)  (14)

Where

UA ^(Balance Point) =UA ^(Occupants) +UA ^(Electric) +UA ^(Solar)  (15)

such that UA^(Occupants), UA^(Electric), and UA^(Solar) respectivelyrepresent the thermal conductivity of internal heating sources,specifically, occupants, electric and solar.

In Equation (14), total thermal conductivity UA^(Total) is fixed at acertain value for a building and is independent of weather conditions;UA^(Total) depends upon the building's efficiency. The component partsof Equation (14), balance point thermal conductivity UA^(Balance) Pointand auxiliary heating thermal conductivity UA^(Auxiliary) Heatinghowever, are allowed to vary with weather conditions. For example, whenthe weather is warm, there may be no auxiliary heating in use and all ofthe thermal conductivity will be allocated to the balance point thermalconductivity UA^(Balance) Point component.

Fuel consumption for heating losses Q^(Losses) can be determined bysubstituting Equation (14) into Equation (6):

Q ^(Losses)=(UA ^(Balance Point) +UA ^(Auxiliary Heating))( T ^(Indoor)−T ^(Outdoor))(H)  (16)

Balance Energy

Heating gains must equal heating losses for the system to balance (step45), as further described infra with reference to FIG. 5 . Heatingenergy balance is represented by setting Equation (7) equal to Equation(16):

Q ^(Gains-Internal) +Q ^(Gains-Auxiliary Heating)=(UA ^(Balance Point)+UA ^(Auxiliary Heating))( T ^(Indoor) −T ^(Outdoor))(H)  (17)

The result can then be divided by (T ^(Indoor)−T ^(Outdoor))(H),assuming that this term is non-zero:

$\begin{matrix}{{{UA}^{{Balance}{Point}} + {UA}^{{Auxiliary}{Heating}}} = \frac{Q^{{Gains} - {Internal}} + Q^{{Gains} - {Auxiliary}{Heating}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (18)\end{matrix}$

Equation (18) expresses energy balance as a combination of bothUA^(Balance Point) and UA^(Auxiliary Heating) FIG. 5 is a flow diagramshowing a routine 60 for balancing energy for use in the method 40 ofFIG. 3 . Equation (18) can be further constrained by requiring that thecorresponding terms on each side of the equation match, which willdivide Equation (18) into a set of two equations:

$\begin{matrix}{{UA}^{{Balance}{Point}} = \frac{Q^{{Gains} - {Internal}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (19)\end{matrix}$ $\begin{matrix}{{UA}^{{Auxiliary}{Heating}} = \frac{Q^{{Gains} - {Auxiliary}{Heating}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (20)\end{matrix}$

Components of UA^(Balance Point)

For clarity, UA^(Balance Point) can be divided into three componentvalues (step 61) by substituting Equation (8) into Equation (19):

$\begin{matrix}{{UA}^{{Balance}{Point}} = \frac{Q^{{Gains} - {Occupants}} + Q^{{Gains} - {Electric}} + Q^{{Gains} - {Solar}}}{\left( {{\overset{\_}{T}}^{Indoor} + {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (21)\end{matrix}$

Since UA^(Balance Point) equals the sum of three component values (asspecified in Equation (15)), Equation (21) can be mathematically limitedby dividing Equation (21) into three equations:

$\begin{matrix}{{UA}^{Occupants} = \frac{Q^{{Gains} - {Occupants}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (22)\end{matrix}$ $\begin{matrix}{{UA}^{Electric} = \frac{Q^{{Gains} - {Electric}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (23)\end{matrix}$ $\begin{matrix}{{UA}^{Solar} = \frac{Q^{{Gains} - {Solar}}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}} & (24)\end{matrix}$

Solutions for Components of UA^(Balance Point) andUA^(Auxiliary Heating)

The preceding equations can be combined to present a set of results withsolutions provided for the four thermal conductivity components asfollows. First, the portion of the balance point thermal conductivityassociated with occupants UA^(Occupants) (step 62) is calculated bysubstituting Equation (9) into Equation (22). Next, the portion of thebalance point thermal conductivity associated with internal electricityconsumption UA^(Electric) (step 63) is calculated by substitutingEquation (10) into Equation (23). The portion of the balance pointthermal conductivity associated with solar gains UA^(Solar) (step 64) isthen calculated by substituting Equation (12) into Equation (24).Finally, thermal conductivity associated with auxiliary heatingUA^(Auxiliary Heating) (step 64) is calculated by substituting Equation(13) into Equation (20).

$\begin{matrix}{{UA}^{Occupants} = \frac{250\left( \overset{\_}{P} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (25)\end{matrix}$ $\begin{matrix}{{UA}^{Electric} = {\frac{\left( \overset{\_}{{Net} + {PV} - {External}} \right)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}\left( \frac{3,412{Btu}}{kWh} \right)}} & (26)\end{matrix}$ $\begin{matrix}{{UA}^{Solar} = {\frac{\left( \overset{\_}{Solar} \right)(W)}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}\left( \frac{3,412{Btu}}{kWh} \right)}} & (27)\end{matrix}$ $\begin{matrix}{{UA}^{{Auxiliary}{Heating}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (28)\end{matrix}$

Determine Fuel Consumption

Referring back to FIG. 3 , Equation (28) can used to derive a solutionto annual (or periodic) heating fuel consumption. First, Equation (14)is solved for UA^(Auxiliary Heating).

UA ^(Auxiliary Heating) =UA ^(Total) −UA ^(Balance Point)  (29)

Equation (29) is then substituted into Equation (28):

$\begin{matrix}{{{UA}^{Total} - {UA}^{{Balance}{Point}}} = \frac{{\overset{\_}{Q}}^{Fuel}\eta^{HVAC}}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (30)\end{matrix}$

Finally, solving Equation (30) for fuel and multiplying by the number ofhours (H) in (or duration of) the time period yields:

$\begin{matrix}{Q^{Fuel} = \frac{\left( {{UA}^{Total} - {UA}^{{Balance}{Point}}} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}} & (31)\end{matrix}$

Equation (31) is valid where UA^(Total)≥UA^(Balance Point). Otherwise,fuel consumption is 0.

Using Equation (31), annual (or periodic) heating fuel consumptionQ^(Fuel) can be determined (step 46). The building's thermalconductivity UA^(Total), if already available through, for instance, theresults of an energy audit, is obtained. Otherwise, UA^(Total) can bedetermined by solving Equations (25) through (28) using historical fuelconsumption data, such as shown, by way of example, in the table of FIG.7 , or by solving Equation (49), as further described infra. UA^(Total)can also be empirically determined with the approach described, forinstance, in commonly-assigned U.S. patent application, entitled “Systemand Method for Empirically Estimating Overall Thermal Performance of aBuilding,” Ser. No. 14/294,087, filed Jun. 2, 2014, pending, thedisclosure of which is incorporated by reference. Other ways todetermine UA^(Total) are possible. UA^(Balance Point) can be determinedby solving Equation (21). The remaining values, average indoortemperature T ^(Indoor) and average outdoor temperature T ^(Outdoor) andη^(HVAC) system efficiency r/HVAC can respectively be obtained fromhistorical weather data and manufacturer specifications.

Practical Considerations

Equation (31) is empowering. Annual heating fuel consumption Q^(Fuel)can be readily determined without encountering the complications ofEquation (1), which is an equation that is difficult to solve due to thenumber and variety of unknown inputs that are required. The implicationsof Equation (31) in consumer decision-making, a general discussion, andsample applications of Equation (31) will now be covered.

Change in Fuel Requirements Associated with Decisions Available toConsumers

Consumers have four decisions available to them that affects theirenergy consumption for heating. FIG. 6 is a process flow diagramshowing, by way of example, consumer heating energy consumption-relateddecision points. These decisions 71 include:

-   -   1. Change the thermal conductivity UA^(Total) by upgrading the        building shell to be more thermally efficient (process 72).        Reduce or change the average indoor temperature by reducing the        thermostat manually, programmatically, or through a “learning”        thermostat (process 73).    -   2. Upgrade the HVAC system to increase efficiency (process 74).    -   3. Increase the solar gain by increasing the effective window        area (process 75).

Other decisions are possible. Here, these four specific options can beevaluated supra by simply taking the derivative of Equation (31) withrespect to a variable of interest. The result for each case is validwhere UA^(Total)≥UA^(Balance Point). Otherwise, fuel consumption is 0.

Changes associated with other internal gains, such as increasingoccupancy, increasing internal electric gains, or increasing solarheating gains, could be calculated using a similar approach.

Change in Thermal Conductivity

A change in thermal conductivity UA^(Total) can affect a change in fuelrequirements. The derivative of Equation (31) is taken with respect tothermal conductivity, which equals the average indoor minus outdoortemperatures times the number of hours divided by HVAC systemefficiency. Note that initial thermal efficiency is irrelevant in theequation. The effect of a change in thermal conductivity UA^(Total)(process 72) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{{dUA}^{Total}} = \frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}} & (32)\end{matrix}$

Change in Average Indoor Temperature

A change in average indoor temperature can also affect a change in fuelrequirements. The derivative of Equation (31) is taken with respect tothe average indoor temperature. Since UA^(Balance Point) is also afunction of average indoor temperature, application of the product ruleis required. After simplifying, the effect of a change in average indoortemperature (process 73) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\overset{\_}{T^{Indoor}}} = {\left( {UA}^{Total} \right)\left( \frac{H}{\eta^{HVAC}} \right)}} & (33)\end{matrix}$

Change in HVAC System Efficiency

As well, a change in HVAC system efficiency can affect a change in fuelrequirements. The derivative of Equation (31) is taken with respect toHVAC system efficiency, which equals current fuel consumption divided byHVAC system efficiency. Note that this term is not linear withefficiency and thus is valid for small values of efficiency changes. Theeffect of a change in fuel requirements relative to the change in HVACsystem efficiency (process 74) can be evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\eta^{HVAC}} = {- {Q^{Fuel}\left( \frac{1}{\eta^{HVAC}} \right)}}} & (34)\end{matrix}$

Change in Solar Gains

An increase in solar gains can be accomplished by increasing theeffective area of south-facing windows. Effective area can be increasedby trimming trees blocking windows, removing screens, cleaning windows,replacing windows with ones that have higher SHGCs, installingadditional windows, or taking similar actions. In this case, thevariable of interest is the effective window area W. The total gain persquare meter of additional effective window area equals the availableresource (kWh/m²) divided by HVAC system efficiency, converted to Btus.The derivative of Equation (31) is taken with respect to effectivewindow area. The effect of an increase in solar gains (process 74) canbe evaluated by solving:

$\begin{matrix}{\frac{{dQ}^{Fuel}}{d\overset{\_}{W}} = {{- \left\lbrack \frac{\left( \overset{\_}{Solar} \right)(H)}{\eta^{HVAC}} \right\rbrack}\left( \frac{3,412{Btu}}{kWh} \right)}} & (35)\end{matrix}$

Discussion

Both Equations (1) and (31) provide ways to calculate fuel consumptionrequirements. The two equations differ in several key ways:

-   -   1. UA^(Total) only occurs in one place in Equation (31), whereas        Equation (1) has multiple indirect and non-linear dependencies        to UA^(Total).    -   2. UA^(Total) is divided into two parts in Equation (31), while        there is only one occurrence of UA^(Total) in Equation (1).    -   3. The concept of balance point thermal conductivity in        Equation (31) replaces the concept of balance point temperature        in Equation (1).    -   4. Heat from occupants, electricity consumption, and solar gains        are grouped together in Equation (31) as internal heating gains,        while these values are treated separately in Equation (1).

Second, Equations (25) through (28) provide empirical methods todetermine both the point at which a building has no auxiliary heatingrequirements and the current thermal conductivity. Equation (1)typically requires a full detailed energy audit to obtain the datarequired to derive thermal conductivity. In contrast, Equations (25)through (28), as applied through the first approach, can substantiallyreduce the scope of an energy audit.

Third, both Equation (4) and Equation (32) provide ways to calculate achange in fuel requirements relative to a change in thermalconductivity. However, these two equations differ in several key ways:

-   -   1. Equation (4) is complex, while Equation (32) is simple.    -   2. Equation (4) depends upon current building thermal        conductivity, balance point temperature, solar savings fraction,        auxiliary heating efficiency, and a variety of other        derivatives. Equation (32) only requires the auxiliary heating        efficiency in terms of building-specific information.

Equation (32) implies that, as long as some fuel is required forauxiliary heating, a reasonable assumption, a change in fuelrequirements will only depend upon average indoor temperature (asapproximated by thermostat setting), average outdoor temperature, thenumber of hours (or other time units) in the (heating) season, and HVACsystem efficiency. Consequently, any building shell (or envelope)investment can be treated as an independent investment. Importantly,Equation (32) does not require specific knowledge about buildingconstruction, age, occupancy, solar gains, internal electric gains, orthe overall thermal conductivity of the building. Only thecharacteristics of the portion of the building that is being replaced,the efficiency of the HVAC system, the indoor temperature (as reflectedby the thermostat setting), the outdoor temperature (based on location),and the length of the winter season are required; knowledge about therest of the building is not required. This simplification is a powerfuland useful result.

Fourth, Equation (33) provides an approach to assessing the impact of achange in indoor temperature, and thus the effect of making a change inthermostat setting. Note that Equation (31) only depends upon theoverall efficiency of the building, that is, the building's totalthermal conductivity UA^(Total), the length of the winter season (innumber of hours or other time units), and the HVAC system efficiency;Equation (31) does not depend upon either the indoor or outdoortemperature.

Equation (31) is useful in assessing claims that are made by HVACmanagement devices, such as the Nest thermostat device, manufactured byNest Labs, Inc., Palo Alto, Calif., or the Lyric thermostat device,manufactured by Honeywell Int'l Inc., Morristown, N.J., or otherso-called “smart” thermostat devices. The fundamental idea behind thesetypes of HVAC management devices is to learn behavioral patterns, sothat consumers can effectively lower (or raise) their average indoortemperatures in the winter (or summer) months without affecting theirpersonal comfort. Here, Equation (31) could be used to estimate thevalue of heating and cooling savings, as well as to verify the consumerbehaviors implied by the new temperature settings.

Balance Point Temperature

Before leaving this section, balance point temperature should briefly bediscussed. The formulation in this first approach does not involvebalance point temperature as an input. A balance point temperature,however, can be calculated to equal the point at which there is no fuelconsumption, such that there are no gains associated with auxiliaryheating (Q^(Gains-Auxiliary Heating) equals 0) and the auxiliary heatingthermal conductivity (UA^(Auxiliary Heating) in Equation (28)) is zero.Inserting these assumptions into Equation (18) and labeling T^(Outdoor)as T^(Balance) Point yields:

Q ^(Gains-Internal) =UA ^(Total)( T ^(Indoor) −T^(Balance Point))(H)  (36)

Equation (36) simplifies to:

$\begin{matrix}{{{\overset{\_}{T}}^{{Balance}{Point}} = {{\overset{\_}{T}}^{Indoor} - \frac{{\overset{\_}{Q}}^{{Gains} - {Internal}}}{{UA}^{Total}}}}{where}{{\overset{\_}{Q}}^{{Gains} - {Internal}} = \frac{Q^{{Gains} - {Internal}}}{H}}} & (37)\end{matrix}$

Equation (37) is identical to Equation (2), except that average valuesare used for indoor temperature T ^(Indoor), balance point temperature T^(Balance Point) and fuel consumption for internal heating gains Q^(Gains-Internal), and that heating gains from occupancy(Q^(Gains-Occupants)) electric (Q^(Gains-Electric)), and solar(Q^(Gains-Solar)) are all included as part of internal heating gains(Q^(Gains-Internal)).

Application: Change in Thermal Conductivity Associated with OneInvestment

An approach to calculating a new value for total thermal conductivity

^(Total) after a series of M changes (or investments) are made to abuilding is described in commonly-assigned U.S. patent application,entitled “System and Method for Interactively Evaluating PersonalEnergy-Related Investments,” Ser. No. 14/294,079, filed Jun. 2, 2014,pending, the disclosure of which is incorporated by reference. Theapproach is summarized therein in Equation (41), which provides:

Total = UA Total + ∑ j = 1 M ( U j - U ^ j ) ⁢ A j + ρ ⁢ c ⁡ ( n - n ^ ) ⁢ V( 38 )

where a caret symbol ({circumflex over ( )}) denotes a new value,infiltration losses are based on the density of air (ρ), specific heatof air (c), number of air changes per hour (n), and volume of air perair change (V). In addition, U^(j) and Û^(j) respectively represent theexisting and proposed U-values of surface j, and A^(j) represents thesurface area of surface j. The volume of the building V can beapproximated by multiplying building square footage by average ceilingheight. The equation, with a slight restatement, equals:

^(Total) =UA ^(Total) +ΔUA ^(Total)  (39)

and

$\begin{matrix}{{\Delta{UA}^{Total}} = {{\sum\limits_{j = 1}^{M}{\left( {U^{j} - {\hat{U}}^{j}} \right)A^{j}}} + {\rho{c\left( {n - \hat{n}} \right)}{V.}}}} & (40)\end{matrix}$

If there is only one investment, the m superscripts can be dropped andthe change in thermal conductivity UA^(Total) equals the area (A) timesthe difference of the inverse of the old and new R-values R and R:

$\begin{matrix}{{\Delta{UA}^{Total}} = {{A\left( {U - \hat{U}} \right)} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}.}}} & (41)\end{matrix}$

Fuel Savings

The fuel savings associated with a change in thermal conductivityUA^(Total) for a single investment equals Equation (41) times (32):

$\begin{matrix}{{\Delta Q^{Fuel}} = {{\Delta{UA}^{Total}\frac{{dQ}^{Fuel}}{{dUA}^{Total}}} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}\frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}}}} & (42)\end{matrix}$

where ΔQ^(Fuel) signifies the change in fuel consumption.

Economic Value

The economic value of the fuel savings (Annual Savings) equals the fuelsavings times the average fuel price (Price) for the building inquestion:

$\begin{matrix}{{{Annual}{Savings}} = {{A\left( {\frac{1}{R} - \frac{1}{\hat{R}}} \right)}\frac{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}({Price})}} & (43)\end{matrix}$ ${where}{{Price} = \left\{ \begin{matrix}{\frac{{Price}^{NG}}{10^{5}}{if}{price}{has}{units}{of}\${per}{therm}} \\{\frac{{Price}^{Electrity}}{10^{5}}{if}{price}{has}{units}{of}\${per}{kWh}}\end{matrix} \right.}$

where Price^(NG) represents the price of natural gas andPrice^(Electricity) represents the price of electricity.

Example

Consider an example. A consumer in Napa, Calif. wants to calculate theannual savings associating with replacing a 20 ft² single-pane windowthat has an R-value of 1 with a high efficiency window that has anR-value of 4. The average temperature in Napa over the 183-day winterperiod (4,392 hours) from October 1 to March 31 is 50° F. The consumersets his thermostat at 68° F., has a 60 percent efficient natural gasheating system, and pays $1 per therm for natural gas. How much moneywill the consumer save per year by making this change?

Putting this information into Equation (43) suggests that he will save$20 per year:

$\begin{matrix}{{{Annual}{Savings}} = {{20\left( {\frac{1}{1} - \frac{1}{4}} \right)\frac{\left( {68 - 50} \right)\left( {4,392} \right)}{0.6}\left( \frac{1}{10^{5}} \right)} = {\$ 20}}} & (44)\end{matrix}$

Application: Validate Building Shell Improvements Savings

Many energy efficiency programs operated by power utilities grapple withthe issue of measurement and evaluation (M&E), particularly with respectto determining whether savings have occurred after building shellimprovements were made. Equations (25) through (28) can be applied tohelp address this issue. These equations can be used to calculate abuilding's total thermal conductivity UA^(Total). This result providesan empirical approach to validating the benefits of building shellinvestments using measured data.

Equations (25) through (28) require the following inputs:

-   -   1) Weather:        -   a) Average outdoor temperature (° F.).        -   b) Average indoor temperature (° F.).        -   c) Average direct solar resource on a vertical, south-facing            surface.    -   2) Fuel and energy:        -   a) Average gross indoor electricity consumption.        -   b) Average natural gas fuel consumption for space heating.        -   c) Average electric fuel consumption for space heating.    -   3) Other inputs:        -   a) Average number of occupants.        -   b) Effective window area.        -   c) HVAC system efficiency.

Weather data can be determined as follows. Indoor temperature can beassumed based on the setting of the thermostat (assuming that thethermostat's setting remained constant throughout the time period), ormeasured and recorded using a device that takes hourly or periodicindoor temperature measurements, such as a Nest thermostat device or aLyric thermostat device, cited supra, or other so-called “smart”thermostat devices. Outdoor temperature and solar resource data can beobtained from a service, such as Solar Anywhere SystemCheck, citedsupra, or the National Weather Service. Other sources of weather dataare possible.

Fuel and energy data can be determined as follows. Monthly utilitybilling records provide natural gas consumption and net electricitydata. Gross indoor electricity consumption can be calculated by addingPV production, whether simulated using, for instance, the Solar AnywhereSystemCheck service, cited supra, or measured directly, and subtractingout external electricity consumption, that is, electricity consumptionfor electric devices that do not deliver all heat that is generated intothe interior of the building. External electricity consumption includeselectric vehicle (EV) charging and electric water heating. Other typesof external electricity consumption are possible. Natural gasconsumption for heating purposes can be estimated by subtractingnon-space heating consumption, which can be estimated, for instance, byexamining summer time consumption using an approach described incommonly-assigned U.S. patent application, entitled “System and Methodfor Facilitating Implementation of Holistic Zero Net EnergyConsumption,” Ser. No. 14/531,933, filed Nov. 3, 2014, pending, thedisclosure of which is incorporated by reference. Other sources of fueland energy data are possible.

Finally, the other inputs can be determined as follows. The averagenumber of occupants can be estimated by the building owner or occupant.Effective window area can be estimated by multiplying actualsouth-facing window area times solar heat gain coefficient (estimated orbased on empirical tests, as further described infra), and HVAC systemefficiency can be estimated (by multiplying reported furnace ratingtimes either estimated or actual duct system efficiency), or can bebased on empirical tests, as further described infra. Other sources ofdata for the other inputs are possible.

Consider an example. FIG. 7 is a table 80 showing, by way of example,data used to calculate thermal conductivity. The data inputs are for asample house in Napa, Calif. based on the winter period of October 1 toMarch 31 for six winter seasons, plus results for a seventh winterseason after many building shell investments were made. (Note thebuilding improvements facilitated a substantial increase in the averageindoor temperature by preventing a major drop in temperature duringnight-time and non-occupied hours.) South-facing windows had aneffective area of 10 m² and the solar heat gain coefficient is estimatedto be 0.25 for an effective window area of 2.5 m². The measured HVACsystem efficiency of 59 percent was based on a reported furnaceefficiency of 80 percent and an energy audit-based duct efficiency of 74percent.

FIG. 8 is a table 90 showing, by way of example, thermal conductivityresults for each season using the data in the table 80 of FIG. 7 asinputs into Equations (25) through (28). Thermal conductivity is inunits of Btu/h-° F. FIG. 9 is a graph 100 showing, by way of example, aplot of the thermal conductivity results in the table 90 of FIG. 8 . Thex-axis represents winter seasons for successive years, each winterseason running from October 1 to March 31. The y-axis represents thermalconductivity. The results from a detailed energy audit, performed inearly 2014, are superimposed on the graph. The energy audit determinedthat the house had a thermal conductivity of 773 Btu/h-° F. The averageresult estimated for the first six seasons was 791 Btu/h-° F. A majoramount of building shell work was performed after the 2013-2014 winterseason, and the results show a 50-percent reduction in heating energyconsumption in the 2014-2015 winter season.

Application: Evaluate Investment Alternatives

The results of this work can be used to evaluate potential investmentalternatives. FIG. 10 is a graph 110 showing, by way of example, anauxiliary heating energy analysis and energy consumption investmentoptions. The x-axis represents total thermal conductivity, UA^(Total) inunits of Btu/hr-° F. The y-axis represents total heating energy. Thegraph presents the analysis of the Napa, Calif. building from theearlier example, supra, using the equations previously discussed. Thethree lowest horizontal bands correspond to the heat provided throughinternal gains 111, including occupants, heat produced by operatingelectric devices, and solar heating. The solid circle 112 represents theinitial situation with respect to heating energy consumption. Thediagonal lines 113 a, 113 b, 113 c represent three alternative heatingsystem efficiencies versus thermal conductivity (shown in the graph asbuilding losses). The horizontal dashed line 114 represents an option toimprove the building shell and the vertical dashed line 115 representsan option to switch to electric resistance heating. The plain circle 116represents the final situation with respect to heating energyconsumption.

Other energy consumption investment options (not depicted) are possible.These options include switching to an electric heat pump, increasingsolar gain through window replacement or tree trimming (this optionwould increase the height of the area in the graph labeled “SolarGains”), or lowering the thermostat setting. These options can becompared using the approach described with reference to Equations (25)through (28) to compare the options in terms of their costs and savings,which will help the homeowner to make a wiser investment.

Second Approach: Time Series Fuel Consumption

The previous section presented an annual fuel consumption model. Thissection presents a detailed time series model. This section alsocompares results from the two methods and provides an example of how toapply the on-site empirical tests.

Building-Specific Parameters

The building temperature model used in this second approach requiresthree building parameters: (1) thermal mass; (2) thermal conductivity;and (3) effective window area. FIG. 11 is a functional block diagramshowing thermal mass, thermal conductivity, and effective window arearelative to a structure 121. By way of introduction, these parameterswill now be discussed.

Thermal Mass (M)

The heat capacity of an object equals the ratio of the amount of heatenergy transferred to the object and the resulting change in theobject's temperature. Heat capacity is also known as “thermalcapacitance” or “thermal mass” (122) when used in reference to abuilding. Thermal mass Q is a property of the mass of a building thatenables the building to store heat, thereby providing “inertia” againsttemperature fluctuations. A building gains thermal mass through the useof building materials with high specific heat capacity and high density,such as concrete, brick, and stone.

The heat capacity is assumed to be constant when the temperature rangeis sufficiently small. Mathematically, this relationship can beexpressed as:

Q _(Δt) =M(T _(t+Δt) ^(Indoor) −T _(t) ^(Indoor))  (45)

where M equals the thermal mass of the building and temperature unitsTare in ° F. Q is typically expressed in Btu or Joules. In that case, Mhas units of Btu/° F. Q can also be divided by 1 kWh/3,412 Btu toconvert to units of kWh/° F.

Thermal Conductivity (UA^(Total))

The building's thermal conductivity UA^(Total) (123) is the amount ofheat that the building gains or losses as a result of conduction andinfiltration. Thermal conductivity UA^(Total) was discussed supra withreference to the first approach for modeling annual heating fuelconsumption.

Effective Window Area (W)

The effective window area (in units of m²) (124), also discussed indetail supra, specifies how much of an available solar resource isabsorbed by the building. Effective window area is the dominant means ofsolar gain in a typical building during the winter and includes theeffect of physical shading, window orientation, and the window's solarheat gain coefficient. In the northern hemisphere, the effective windowarea is multiplied by the available average direct irradiance on avertical, south-facing surface (kW/m²), times the amount of time (H) toresult in the kWh obtained from the windows.

Energy Gain or Loss

The amount of heat transferred to or extracted from a building (Q) overa time period of Δt is based on a number of factors, including:

-   -   1) Loss (or gain if outdoor temperature exceeds indoor        temperature) due to conduction and infiltration and the        differential between the indoor and outdoor temperatures.    -   2) Gain associated with:        -   a) Occupancy and heat given off by people.        -   b) Heat produced by consuming electricity inside the            building.        -   c) Solar radiation.        -   d) Auxiliary heating.

Mathematically, Q can be expressed as:

$\begin{matrix}{Q_{\Delta t} = {\left\lbrack {\overset{\overset{{Envelope}{Gain}{or}{Loss}}{︷}}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} + \overset{\overset{{Occupancy}{Gain}}{︷}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}{Electric}{Gain}}{︷}}{\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + \overset{\overset{{Solar}{Gain}}{︷}}{W{\overset{\_}{Solar}\left( \frac{3,412{Btu}}{1{kWh}} \right)}} + \overset{\overset{{Auxliary}{Heating}{Gain}}{︷}}{R^{Furnace}\eta^{HVAC}\overset{\_}{Status}}} \right\rbrack\Delta t}} & (46)\end{matrix}$

where:

-   -   Except as noted otherwise, the bars over the variable names        represent the average value over Δt hours, that is, the duration        of the applicable empirical test. For instance, T ^(Outdoor)        represents the average outdoor temperature between the time        interval of t and t+Δt.    -   UA^(Total) is the thermal conductivity (in units of Btu/hour-°        F.).    -   W is the effective window area (in units of m²).    -   Occupancy Gain is based on the average number of people (P) in        the building during the applicable empirical test (and the heat        produced by those people). The average person is assumed to        produce 250 Btu/hour.    -   Internal Electric Gain is based on heat produced by indoor        electricity consumption (Electric), as averaged over the        applicable empirical test, but excludes electricity for purposes        that do not produce heat inside the building, for instance,        electric hot water heating where the hot water is discarded down        the drain, or where there is no heat produced inside the        building, such as is the case with EV charging.    -   Solar Gain is based on the average available normalized solar        irradiance (Solar) during the applicable empirical test (with        units of kW/m²). This value is the irradiance on a vertical        surface to estimate solar received on windows; global horizontal        irradiance (GHI) can be used as a proxy for this number when W        is allowed to change on a monthly basis.    -   Auxiliary Heating Gain is based on the rating of the furnace (R        in Btu), HVAC system efficiency (η^(HVAC), including both        furnace and delivery system efficiency), and average furnace        operation status (Status) during the empirical test, a time        series value that is either off (0 percent) or on (100 percent).

Energy Balance

Equation (45) reflects the change in energy over a time period andequals the product of the temperature change and the building's thermalmass. Equation (46) reflects the net gain in energy over a time periodassociated with the various component sources. Equation (45) can be setto equal Equation (46), since the results of both equations equal thesame quantity and have the same units (Btu). Thus, the total heat changeof a building will equal the sum of the individual heat gain/losscomponents:

$\begin{matrix}{\overset{\overset{{Total}{Heat}{Change}}{︷}}{M\left( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} \right)} = {\left\lbrack {\overset{\overset{{Envelope}{Gain}{or}{Loss}}{︷}}{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} + \overset{\overset{{Occupancy}{Gain}}{︷}}{(250)\overset{\_}{P}} + \overset{\overset{{Internal}{Electric}{Gain}}{︷}}{\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + \overset{\overset{{Solar}{Gain}}{︷}}{W{\overset{\_}{Solar}\left( \frac{3,412{Btu}}{1{kWh}} \right)}} + \overset{\overset{{Auxliary}{Heating}{Gain}}{︷}}{R^{Furnace}\eta^{HVAC}\overset{\_}{Status}}} \right\rbrack\Delta t}} & (47)\end{matrix}$

Equation (47) can be used for several purposes. FIG. 12 is a flowdiagram showing a computer-implemented method 130 for modeling intervalbuilding heating energy consumption in accordance with a furtherembodiment. Execution of the software can be performed with theassistance of a computer system, such as further described infra withreference to FIG. 22 , as a series of process or method modules orsteps.

As a single equation, Equation (47) is potentially very useful, despitehaving five unknown parameters. In this second approach, the unknownparameters are solved by performing a series of short duration empiricaltests (step 131), as further described infra with reference to FIG. 14 .Once the values of the unknown parameters are found, a time series ofindoor temperature data can be constructed (step 132), which will thenallow annual fuel consumption to be calculated (step 133) and maximumindoor temperature to be found (step 134). The short duration tests willfirst be discussed.

Empirically Determine Building- and Equipment-Specific Parameters UsingShort Duration Tests

A series of tests can be used to iteratively solve Equation (47) toobtain the values of the unknown parameters by ensuring that theportions of Equation (47) with the unknown parameters are equal to zero.FIG. 13 is a table 140 showing the characteristics of empirical testsused to solve for the five unknown parameters in Equation (47). Theempirical test characteristics are used in a series ofsequentially-performed short duration tests; each test builds on thefindings of earlier tests to replace unknown parameters with foundvalues.

FIG. 14 is a flow diagram showing a routine 150 for empiricallydetermining building- and equipment-specific parameters using shortduration tests for use in the method 130 of FIG. 12 . The approach is torun a serialized series of empirical tests. The first test solves forthe building's total thermal conductivity (UA^(Total)) (step 151). Thesecond test uses the empirically-derived value for UA^(Total) to solvefor the building's thermal mass (M) (step 152). The third test uses bothof these results, thermal conductivity and thermal mass, to find thebuilding's effective window area (W) (step 153). Finally, the fourthtest uses the previous three test results to determine the overall HVACsystem efficiency (step 145). Consider how to perform each of thesetests.

Test 1: Building Thermal Conductivity (UA^(Total))

The first step is to find the building's total thermal conductivity(UA^(Total)) (step 151). Referring back to the table in FIG. 13 , thisshort-duration test occurs at night (to avoid any solar gain) with theHVAC system off (to avoid any gain from the HVAC system), and by havingthe indoor temperature the same at the beginning and the ending of thetest by operating an electric controllable interior heat source, such asportable electric space heaters that operate at 100% efficiency, so thatthere is no change in the building temperature's at the beginning and atthe ending of the test. Thus, the interior heart source must havesufficient heating capacity to maintain the building's temperaturestate. Ideally, the indoor temperature would also remain constant toavoid any potential concerns with thermal time lags.

These assumptions are input into Equation (47):

$\begin{matrix}{{M(0)} = {\left\lbrack {{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} + {(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {{W(0)}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {R^{Furnace}{\eta^{HVAC}(0)}}} \right\rbrack\Delta t}} & (48)\end{matrix}$

The portions of Equation (48) that contain four of the five unknownparameters now reduce to zero. The result can be solved for UA^(Total):

$\begin{matrix}{{UA}^{Total} = \frac{\left\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)}} \right\rbrack}{\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}} & (49)\end{matrix}$

where T ^(Indoor) represents the average indoor temperature during theempirical test, T ^(Outdoor) represents the average outdoor temperatureduring the empirical test, P represents the average number of occupantsduring the empirical test, and Electric represents average indoorelectricity consumption during the empirical test.

Equation (49) implies that the building's thermal conductivity can bedetermined from this test based on average number of occupants, averagepower consumption, average indoor temperature, and average outdoortemperature.

Test 2: Building Thermal Mass (M)

The second step is to find the building's thermal mass (M) (step 152).This step is accomplished by constructing a test that guarantees M isspecifically non-zero since UA^(Total) is known based on the results ofthe first test. This second test is also run at night, so that there isno solar gain, which also guarantees that the starting and the endingindoor temperatures are not the same, that is, T_(t+Δt) ^(Indoor)≠T_(t)^(Indoor), respectively at the outset and conclusion of the test by notoperating the HVAC system. These assumptions are input into Equation(47) and solving yields a solution for M:

$\begin{matrix}{M = {\left\lbrack \frac{{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} + {(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)}}{\left( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} \right)} \right\rbrack\Delta t}} & (50)\end{matrix}$

where UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature during the empirical test, T^(Outdoor) represents the average outdoor temperature during theempirical test, P represents the average number of occupants during theempirical test, Electric represents average indoor electricityconsumption during the empirical test, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, T_(t) ^(Indoor) represents the starting indoor temperature,and T_(t+Δt) ^(Indoor)≠T_(t) ^(Indoor).

Test 3: Building Effective Window Area (W)

The third step to find the building's effective window area (W) (step153) requires constructing a test that guarantees that solar gain isnon-zero. This test is performed during the day with the HVAC systemturned off. Solving for W yields:

$\begin{matrix}{W = {\left\{ {\left\lbrack \frac{M\left( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} \right)}{3,412\Delta t} \right\rbrack + \frac{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}{3,412} - \frac{(250)\overset{\_}{P}}{3,412} - \overset{\_}{Electric}} \right\}\left\lbrack \frac{1}{\overset{\_}{Solar}} \right\rbrack}} & (51)\end{matrix}$

where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T_(t) ^(Indoor) represents the starting indoortemperature, UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature, T ^(Outdoor) represents theaverage outdoor temperature, P represents the average number ofoccupants during the empirical test, Electric represents averageelectricity consumption during the empirical test, and Solar representsthe average solar energy produced during the empirical test.

Test 4: HVAC System Efficiency (η^(Furnace)η^(Delivery))

The fourth step determines the HVAC system efficiency (step 154). TotalHVAC system efficiency is the product of the furnace efficiency and theefficiency of the delivery system, that is, the duct work and heatdistribution system. While these two terms are often solved separately,the product of the two terms is most relevant to building temperaturemodeling. This test is best performed at night, so as to eliminate solargain. Thus:

$\begin{matrix}{\eta^{HVAC} = {\left\lbrack {\frac{M\left( {T_{t + {\Delta t}}^{Indoor} - T_{t}^{Indoor}} \right)}{\Delta t} - {{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)} - {(250)\overset{\_}{P}} - {\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)}} \right\rbrack\left\lbrack \frac{1}{R^{Furnace}\overset{\_}{Status}} \right\rbrack}} & (52)\end{matrix}$

where M represents the thermal mass, t represents the time at thebeginning of the empirical test, Δt represents the duration of theempirical test, T_(t+Δt) ^(Indoor) represents the ending indoortemperature, and T_(t) ^(Indoor) represents the starting indoortemperature, UA^(Total) represents the thermal conductivity, T ^(Indoor)represents the average indoor temperature, T ^(Outdoor) represents theaverage outdoor temperature, P represents the average number ofoccupants during the empirical test, Electric represents averageelectricity consumption during the empirical test, Status represents theaverage furnace operation status, and R^(Furnace) represents the ratingof the furnace.

Note that HVAC duct efficiency can be determined without performing aduct leakage test if the generation efficiency of the furnace is known.This observation usefully provides an empirical method to measure ductefficiency without having to perform a duct leakage test.

Time Series Indoor Temperature Data

The previous subsection described how to perform a series of empiricalshort duration tests to determine the unknown parameters in Equation(47). Commonly-assigned U.S. patent application Ser. No. 14/531,933,cited supra, describes how a building's UA^(Total) can be combined withhistorical fuel consumption data to estimate the benefit of improvementsto a building. While useful, estimating the benefit requires measuredtime series fuel consumption and HVAC system efficiency data. Equation(47), though, can be used to perform the same analysis without the needfor historical fuel consumption data.

Referring back to FIG. 12 , Equation (47) can be used to construct timeseries indoor temperature data (step 132) by making an approximation.Let the time period (Δt) be short (an hour or less), so that the averagevalues are approximately equal to the value at the beginning of the timeperiod, that is, assume T^(Outdoor) ≈T^(Outdoor). The average values inEquation (47) can be replaced with time-specific subscripted values andsolved to yield the final indoor temperature.

$\begin{matrix}{T_{t + {\Delta t}}^{Indoor} = {T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {{Electric}_{t}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {{WSolar}_{t}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {R^{Furnace}\eta^{HVAC}{Status}_{t}}} \right\rbrack}\Delta t}}} & (53)\end{matrix}$

Once T_(t+Δt) ^(Indoor) is known, Equation (53) can be used to solve forT_(t+2Δt) ^(Indoor) and so on.

Importantly, Equation (53) can be used to iteratively construct indoorbuilding temperature time series data with no specific information aboutthe building's construction, age, configuration, number of stories, andso forth. Equation (53) only requires general weather datasets (outdoortemperature and irradiance) and building-specific parameters. Thecontrol variable in Equation (53) is the fuel required to deliver theauxiliary heat at time t, as represented in the Status variable, thatis, at each time increment, a decision is made whether to run the HVACsystem.

Annual Fuel Consumption

Equation (47) can also be used to calculate annual fuel consumption(step 133) by letting Δt equal the number of hours in the entire timeperiod (H) (and not the duration of the applicable empirical test),rather than making Δt very short (such as an hour, as used in anapplicable empirical test). The indoor temperature at the start and theend of the season can be assumed to be the same or, alternatively, thetotal heat change term on the left side of the equation can be assumedto be very small and set equal to zero. Rearranging the equationprovides:

$\begin{matrix}{{R^{Furnace}\eta^{HVAC}{\overset{\_}{Status}(H)}} = {{{{UA}^{Total}\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)}(H)} - {{\left\lbrack {{(250)\overset{\_}{P}} + {\overset{\_}{Electric}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {W{\overset{\_}{Solar}\left( \frac{3,412{Btu}}{1{kWh}} \right)}}} \right\rbrack(H)}}}} & (54)\end{matrix}$

The left side of the equation, R^(Furnace)η^(HVAC) Status(H), equalstotal fuel consumption. The second term in the right side of theequation equals the internal gains described in Equation (21). WhenEquation (21) is rearranged, UA^(Balance Point) (T ^(Indoor)−T^(Outdoor))(H)=Q^(Gains-Occupants)+Q^(Gains-Electric)+Q^(Gains-Solar).After substituting into Equation (54), the total fuel consumption usingthe time series method produces a result that is identical to the oneproduced using the annual method described in Equation (31).

$\begin{matrix}{Q^{Fuel} = \frac{\left( {{UA}^{Total} - {UA}^{{Balance}{Point}}} \right)\left( {{\overset{\_}{T}}^{Indoor} - {\overset{\_}{T}}^{Outdoor}} \right)(H)}{\eta^{HVAC}}} & (55)\end{matrix}$

Maximum Indoor Temperature

Allowing consumers to limit the maximum indoor temperature to some valuecan be useful from a personal physical comfort perspective. The limit ofmaximum indoor temperature (step 134) can be obtained by taking theminimum of T_(t+Δt) ^(Indoor) and T^(Indoor-Max), the maximum indoortemperature recorded for the building during the heating season. Therecan be some divergence between the annual and detailed time seriesmethods when the thermal mass of the building is unable to absorb excessheat, which can then be used at a later time. Equation (53) becomesEquation (56) when the minimum is applied.

$\begin{matrix}{T_{t + {\Delta t}}^{Indoor} = {{Min}\left\{ {T^{{Indoor} - \max},{T_{t}^{Indoor} + {{\left\lbrack \frac{1}{M} \right\rbrack\left\lbrack {{{UA}^{Total}\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {{Electric}_{t}{{\left( \frac{3,412{Btu}}{1{kWh}} \right) + {{WSolar}_{t}\left( \frac{3,412{Btu}}{1{kWh}} \right)} + {R^{Furnace}\eta^{HVAC}{Status}_{t}}}}}} \right\rbrack}\Delta t}}} \right\}}} & (56)\end{matrix}$

Comparison to Annual Method (First Approach)

Two different approaches to calculating annual fuel consumption aredescribed herein. The first approach, per Equation (31), is asingle-line equation that requires six inputs. The second approach, perEquation (56), constructs a time series dataset of indoor temperatureand HVAC system status. The second approach considers all of theparameters that are indirectly incorporated into the first approach. Thesecond approach also includes the building's thermal mass and thespecified maximum indoor temperature, and requires hourly time seriesdata for the following variables: outdoor temperature, solar resource,non-HVAC electricity consumption, and occupancy.

Both approaches were applied to the exemplary case, discussed supra, forthe sample house in Napa, Calif. Thermal mass was 13,648 Btu/° F. andthe maximum temperature was set at 72° F. The auxiliary heating energyrequirements predicted by the two approaches was then compared. FIG. 15is a graph 160 showing, by way of example, a comparison of auxiliaryheating energy requirements determined by the hourly approach versus theannual approach. The x-axis represents total thermal conductivity,UA^(Total) in units of Btu/hr-° F. The y-axis represents total heatingenergy. FIG. 15 uses the same format as the graph in FIG. 10 by applyinga range of scenarios. The red line in the graph corresponds to theresults of the hourly method. The dashed black line in the graphcorresponds to the annual method. The graph suggests that results areessentially identical, except when the building losses are very low andsome of the internal gains are lost due to house overheating, which isprevented in the hourly method, but not in the annual method.

The analysis was repeated using a range of scenarios with similarresults. FIG. 16 is a graph 170 showing, by way of example, a comparisonof auxiliary heating energy requirements with the allowable indoortemperature limited to 2° F. above desired temperature of 68° F. Here,the only cases that found any meaningful divergence occurred when themaximum house temperature was very close to the desired indoortemperature. FIG. 17 is a graph 180 showing, by way of example, acomparison of auxiliary heating energy requirements with the size ofeffective window area tripled from 2.5 m² to 7.5 m². Here, internalgains were large by tripling solar gains and there was insufficientthermal mass to provide storage capacity to retain the gains.

The conclusion is that both approaches yield essentially identicalresults, except for cases when the house has inadequate thermal mass toretain internal gains (occupancy, electric, and solar).

Example

How to perform the tests described supra using measured data can beillustrated through an example. These tests were performed between 9 PMon Jan. 29, 2015 to 6 AM on Jan. 31, 2015 on a 35 year-old, 3,000 ft²house in Napa, Calif. This time period was selected to show that all ofthe tests could be performed in less than a day-and-a-half. In addition,the difference between indoor and outdoor temperatures was not extreme,making for a more challenging situation to accurately perform the tests.

FIG. 18 is a table 190 showing, by way of example, test data. The subcolumns listed under “Data” present measured hourly indoor and outdoortemperatures, direct irradiance on a vertical south-facing surface(VDI), electricity consumption that resulted in indoor heat, and averageoccupancy. Electric space heaters were used to heat the house and theHVAC system was not operated. The first three short-duration tests,described supra, were applied to this data. The specific data used arehighlighted in gray. FIG. 19 is a table 200 showing, by way of example,the statistics performed on the data in the table 190 of FIG. 18required to calculate the three test parameters. UA^(Total) wascalculated using the data in the table of FIG. 10 and Equation (49).Thermal Mass (M) was calculated using UA^(Total), the data in the tableof FIG. 10 , and Equation (50). Effective Window Area (W) was calculatedusing UA^(Total), M, the data in the table of FIG. 10 , and Equation(51).

These test parameters, plus a furnace rating of 100,000 Btu/hour andassumed efficiency of 56%, can be used to generate the end-of-periodindoor temperature by substituting them into Equation (53) to yield:

$\begin{matrix}{T_{t + {\Delta t}}^{Indoor} = {T_{t}^{Indoor} + {{\left\lbrack \frac{1}{18,084} \right\rbrack\left\lbrack {{429\left( {T_{t}^{Outdoor} - T_{t}^{Indoor}} \right)} + {(250)P_{t}} + {3412{Electric}_{t}} + {11,600{Solar}_{t}} + {\left( {100,000} \right)(0.56){Status}_{t}}} \right\rbrack}\Delta t}}} & (57)\end{matrix}$

Indoor temperatures were simulated using Equation (57) and the requiredmeasured time series input datasets. Indoor temperature was measuredfrom Dec. 15, 2014 to Jan. 31, 2015 for the test location in Napa,Calif. The temperatures were measured every minute on the first andsecond floors of the middle of the house and averaged. FIG. 20 is agraph 210 showing, by way of example, hourly indoor (measured andsimulated) and outdoor (measured) temperatures. FIG. 21 is a graph 220showing, by way of example, simulated versus measured hourly temperaturedelta (indoor minus outdoor). FIG. 20 and FIG. 21 suggest that thecalibrated model is a good representation of actual temperatures.

Energy Consumption Modeling System

Modeling energy consumption for heating (or cooling) on an annual (orperiodic) basis, as described supra with reference FIG. 3 , and on anhourly (or interval) basis, as described supra beginning with referenceto FIG. 12 , can be performed with the assistance of a computer, orthrough the use of hardware tailored to the purpose. FIG. 22 is a blockdiagram showing a computer-implemented system 230 for modeling buildingheating energy consumption in accordance with one embodiment. A computersystem 231, such as a personal, notebook, or tablet computer, as well asa smartphone or programmable mobile device, can be programmed to executesoftware programs 232 that operate autonomously or under user control,as provided through user interfacing means, such as a monitor, keyboard,and mouse. The computer system 231 includes hardware componentsconventionally found in a general purpose programmable computing device,such as a central processing unit, memory, input/output ports, networkinterface, and non-volatile storage, and execute the software programs232, as structured into routines, functions, and modules. In addition,other configurations of computational resources, whether provided as adedicated system or arranged in client-server or peer-to-peertopologies, and including unitary or distributed processing,communications, storage, and user interfacing, are possible.

In one embodiment, to perform the first approach, the computer system231 needs data on heating losses and heating gains, with the latterseparated into internal heating gains (occupant, electric, and solar)and auxiliary heating gains. The computer system 231 may be remotelyinterfaced with a server 240 operated by a power utility or otherutility service provider 241 over a wide area network 239, such as theInternet, from which fuel purchase data 242 can be retrieved.Optionally, the computer system 231 may also monitor electricity 234 andother metered fuel consumption, where the meter is able to externallyinterface to a remote machine, as well as monitor on-site powergeneration, such as generated by a photovoltaic system 235. Themonitored fuel consumption and power generation data can be used tocreate the electricity and heating fuel consumption data and historicalsolar resource and weather data. The computer system 231 then executes asoftware program 232 to determine annual (or periodic) heating fuelconsumption 244 based on the empirical approach described supra withreference to FIG. 3 .

In a further embodiment, to assist with the empirical tests performed inthe second approach, the computer system 231 can be remotely interfacedto a heating source 236 and a thermometer 237 inside a building 233 thatis being analytically evaluated for thermal performance, thermal mass,effective window area, and HVAC system efficiency. In a furtherembodiment, the computer system 231 also remotely interfaces to athermometer 238 outside the building 163, or to a remote data sourcethat can provide the outdoor temperature. The computer system 231 cancontrol the heating source 236 and read temperature measurements fromthe thermometer 237 throughout the short-duration empirical tests. In afurther embodiment, a cooling source (not shown) can be used in place ofor in addition to the heating source 236. The computer system 231 thenexecutes a software program 232 to determine hourly (or interval)heating fuel consumption 244 based on the empirical approach describedsupra with reference to FIG. 12 .

Applications

The two approaches to estimating energy consumption for heating (orcooling), hourly and annual, provide a powerful set of tools that can beused in various applications. A non-exhaustive list of potentialapplications will now be discussed. Still other potential applicationsare possible.

Application to Homeowners

Both of the approaches, annual (or periodic) and hourly (or interval),reformulate fundamental building heating (and cooling) analysis in amanner that can divide a building's thermal conductivity into two parts,one part associated with the balance point resulting from internal gainsand one part associated with auxiliary heating requirements. These twoparts provide that:

-   -   Consumers can compare their house to their neighbors' houses on        both a total thermal conductivity UA^(Total) basis and on a        balance point per square foot basis. These two numbers, total        thermal conductivity UA^(Total) and balance point per square        foot, can characterize how well their house is doing compared to        their neighbors' houses. The comparison could also be performed        on a neighborhood- or city-wide basis, or between comparably        built houses in a subdivision. Other types of comparisons are        possible.    -   As strongly implied by the empirical analyses discussed supra,        heater size can be significantly reduced as the interior        temperature of a house approaches its balance point temperature.        While useful from a capital cost perspective, a heater that was        sized based on this implication may be slow to heat up the house        and could require long lead times to anticipate heating needs.        Temperature and solar forecasts can be used to operate the        heater by application of the two approaches described supra, so        as to optimize operation and minimize consumption. For example,        if the building owner or occupant knew that the sun was going to        start adding a lot of heat to the building in a few hours, he        may choose to not have the heater turn on. Alternatively, if the        consumer was using a heater with a low power rating, he would        know when to turn the heater off to achieve desired preferences.

Application to Building Shell Investment Valuation

The economic value of heating (and cooling) energy savings associatedwith any building shell improvement in any building has been shown to beindependent of building type, age, occupancy, efficiency level, usagetype, amount of internal electric gains, or amount solar gains, providedthat fuel has been consumed at some point for auxiliary heating. Asindicated by Equation (43), the only information required to calculatesavings includes the number of hours that define the winter season;average indoor temperature; average outdoor temperature; the building'sHVAC system efficiency (or coefficient of performance for heat pumpsystems); the area of the existing portion of the building to beupgraded; the R-value of the new and existing materials; and the averageprice of energy, that is, heating fuel. This finding means, for example,that a high efficiency window replacing similar low efficiency windowsin two different buildings in the same geographical location for twodifferent customer types, for instance, a residential customer versus anindustrial customer, has the same economic value, as long as the HVACsystem efficiencies and fuel prices are the same for these two differentcustomers.

This finding vastly simplifies the process of analyzing the value ofbuilding shell investments by fundamentally altering how the analysisneeds to be performed. Rather than requiring a full energy audit-styleanalysis of the building to assess any the costs and benefits of aparticular energy efficiency investment, only the investment ofinterest, the building's HVAC system efficiency, and the price and typeof fuel being saved are required.

As a result, the analysis of a building shell investment becomes muchmore like that of an appliance purchase, where the energy savings, forexample, equals the consumption of the old refrigerator minus the costof the new refrigerator, thereby avoiding the costs of a whole housebuilding analysis. Thus, a consumer can readily determine whether anacceptable return on investment will be realized in terms of costsversus likely energy savings. This result could be used in a variety ofplaces:

-   -   Direct display of economic impact in ecommerce sites. A Web        service that estimates economic value can be made available to        Web sites where consumers purchase building shell replacements.        The consumer would select the product they are purchasing, for        instance, a specific window, and would either specify the        product that they are replacing or a typical value can be        provided. This information would be submitted to the Web        service, which would then return an estimate of savings using        the input parameters described supra.    -   Tools for salespeople at retail and online establishments.    -   Tools for mobile or door-to-door sales people.    -   Tools to support energy auditors for immediate economic        assessment of audit findings. For example, a picture of a        specific portion of a house can be taken and the dollar value of        addressing problems can be attached.    -   Have a document with virtual sticky tabs that show economics of        exact value for each portion of the house. The document could be        used by energy auditors and other interested parties.    -   Available to companies interacting with new building purchasers        to interactively allow them to understand the effects of        different building choices from an economic (and environmental)        perspective using a computer program or Internet-based tool.    -   Enable real estate agents working with customers at the time of        a new home purchase to quantify the value of upgrades to the        building at the time of purchase.    -   Tools to simplify the optimization problem because most parts of        the problem are separable and simply require a rank ordering of        cost-benefit analysis of the various measures and do not require        detailed computer models that applied to specific houses.    -   The time to fix the insulation and ventilation in a homeowner's        attic is when during reroofing. This result could be integrated        into the roofing quoting tools.    -   Incorporated into a holistic zero net energy analysis computer        program or Web site to take an existing building to zero net        consumption.    -   Integration into tools for architects, builders, designers for        new construction or retrofit. Size building features or HVAC        system. More windows or less windows will affect HVAC system        size.

Application to Thermal Conductivity Analysis

A building's thermal conductivity can be characterized using onlymeasured utility billing data (natural gas and electricity consumption)and assumptions about effective window area, HVAC system efficiency andaverage indoor building temperature. This test could be used as follows:

-   -   Utilities lack direct methods to measure the energy savings        associated with building shell improvements. Use this test to        provide a method for electric utilities to validate energy        efficiency investments for their energy efficiency programs        without requiring an on-site visit or the typical detailed        energy audit. This method would help to address the measurement        and evaluation (M&E) issues currently associated with energy        efficiency programs.    -   HVAC companies could efficiently size HVAC systems based on        empirical results, rather than performing Manual J calculations        or using rules of thumb. This test could save customers money        because Manual J calculations require a detailed energy audit.        This test could also save customers capital costs since rules of        thumb typically oversize HVAC systems, particularly for        residential customers, by a significant margin.    -   A company could work with utilities (who have energy efficiency        goals) and real estate agents (who interact with customers when        the home is purchased) to identify and target inefficient homes        that could be upgraded at the time between sale and occupancy.        This approach greatly reduces the cost of the analysis, and the        unoccupied home offers an ideal time to perform upgrades without        any inconvenience to the homeowners.    -   Goals could be set for consumers to reduce a building's heating        needs to the point where a new HVAC system is avoided        altogether, thus saving the consumer a significant capital cost.

Application to Building Performance Studies

A building's performance can be fully characterized in terms of fourparameters using a suite of short-duration (several day) tests. The fourparameters include thermal conductivity, that is, heat losses, thermalmass, effective window area, and HVAC system efficiency. An assumptionis made about average indoor building temperature. These (or theprevious) characterizations could be used as follows:

-   -   Utilities could identify potential targets for building shell        investments using only utility billing data. Buildings could be        identified in a two-step process. First, thermal conductivity        can be calculated using only electric and natural gas billing        data, making the required assumptions presented supra. Buildings        that pass this screen could be the focus of a follow-up,        on-site, short-duration test.    -   The results from this test suite can be used to generate        detailed time series fuel consumption data (either natural gas        or electricity). This data can be combined with an economic        analysis tool, such as the PowerBill service        (http://www.cleanpower.com/products/powerbill/), a software        service offered by Clean Power Research, L.L.C., Napa, Calif.,        to calculate the economic impacts of the changes using detailed,        time-of-use rate structures.

Application to “Smart” Thermostat Users

The results from the short-duration tests, as described supra withreference to FIG. 4 , could be combined with measured indoor buildingtemperature data collected using an Internet-accessible thermostat, suchas a Nest thermostat device or a Lyric thermostat device, cited supra,or other so-called “smart” thermostat devices, thereby avoiding havingto make assumptions about indoor building temperature. The buildingcharacterization parameters could then be combined with energyinvestment alternatives to educate consumers about the energy, economic,and environmental benefits associated with proposed purchases.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A system for balance-point-thermal-conductivity-based building analysis with the aid of a digital computer, comprising: a computer comprising a processor configured to execute code stored in a memory, the computer configured to: obtain a total thermal conductivity of a building; identify a balance point thermal conductivity of the building based on a balance point up to which the building can be thermally sustained using only internal heating period for a time period; divide the balance point thermal conductivity by an area of the building to obtain the balance point thermal conductivity per unit of the area; obtain a further balance point thermal conductivity per the unit of a further area of at least one further building and a further total thermal conductivity of the at least one further building; and compare the balance point thermal conductivity per unit of the area of the building to the further balance point thermal conductivity per the unit of the further area of the at least one further building and compare the total thermal conductivity to the further total conductivity of the at least one building.
 2. A system according to claim 1, the computer further configured to remotely control a heating source inside the building to obtain the thermal conductivity.
 3. A system according to claim 2, wherein the thermal conductivity is obtained using an empirical test conducted using the heating source.
 4. A system according to claim 1, the computer further configured to model at least one change to the building, wherein the change is performed based on the comparison.
 5. A system according to claim 1, wherein the units are square feet.
 6. A system according to claim 1, wherein the at least one further building is neighboring the building.
 7. A system according to claim 1, wherein the at least one further building is in a same city as the building.
 8. A system according to claim 1, the computer further configured to: identify a temperature difference between an average temperature outside and an average temperature inside the building over a time period for the time period; identify internal heating gains within the building over the time period; and find the balance point thermal conductivity as a function of the internal heating gains over the temperature difference and duration of the heating time period.
 9. A system according to claim 1, wherein the internal heating gains are identified using occupant heating gains for the building, heating gains produced by operation of electric devices in the building, and solar heating gains for the building.
 10. A system according to claim 1, wherein the total thermal conductivity comprises the balance point thermal conductivity and thermal conductivity for auxiliary heating.
 11. A method for balance-point-thermal-conductivity-based building analysis with the aid of a digital computer, comprising: obtaining by a computer, the computer comprising a processor configured to execute code stored in a memory, a total thermal conductivity of a building; identifying by the computer a balance point thermal conductivity of the building based on a balance point up to which the building can be thermally sustained using only internal heating period for a time period; dividing by the computer the balance point thermal conductivity by an area of the building to obtain the balance point thermal conductivity per unit of the area; obtaining by the computer a further balance point thermal conductivity per the unit of a further area of at least one further building and a further total thermal conductivity of the at least one further building; and comparing by the computer the balance point thermal conductivity per unit of the area of the building to the further balance point thermal conductivity per the unit of the further area of the at least one further building and comparing by the computer the total thermal conductivity to the further total conductivity of the at least one building.
 12. A method according to claim 11, further comprising remotely controlling a heating source inside the building to obtain the thermal conductivity.
 13. A method according to claim 12, wherein the thermal conductivity is obtained using an empirical test conducted using the heating source.
 14. A method according to claim 11, further comprising modeling at least one change to the building, wherein the change is performed based on the comparison.
 15. A method according to claim 11, wherein the units are square feet.
 16. A method according to claim 11, wherein the at least one further building is neighboring the building.
 17. A method according to claim 11, wherein the at least one further building is in a same city as the building.
 18. A method according to claim 11, further comprising: identify a temperature difference between an average temperature outside and an average temperature inside the building over a time period for the time period; identify internal heating gains within the building over the time period; and find the balance point thermal conductivity as a function of the internal heating gains over the temperature difference and duration of the heating time period.
 19. A method according to claim 11, wherein the internal heating gains are identified using occupant heating gains for the building, heating gains produced by operation of electric devices in the building, and solar heating gains for the building.
 20. A method according to claim 11, wherein the total thermal conductivity comprises the balance point thermal conductivity and thermal conductivity for auxiliary heating. 